/*
U4(2) as 6 x 6 matrices over Z.
Representation 6.
Schur index 1.

SEED:
Nonzero v fixed by <y*x*y^2*x,x*y*x*y*x*y^-1*x> = 2^4:A5.
v has 1 x 27 = 27 images under G; <v> has 27 images under G.
BASIS:
v, v*x, v*y, v*x*y, v*y^2, v*x*y*x.

Possible matrix entries are in {-1,0,1}.

Average number of nonzero entries for any element of the group:
16  (exactly 16; about 44.444%).

Entry   Av/Mat  %Av/Mat
    0    20      55.556 [55+5/9]
   ±1    16      44.444 [44+4/9]
    1     8      22.222 [22+2/9]
   -1     8      22.222 [22+2/9]


LATTICE INFO:
Name: E6*.
Minimum: 4.
Kissing number: 54.
Theta series: 1 + 54*q^4 + 72*q^6 + 432*q^10 + O(q^12).


p-MODULAR REDUCTIONS:
p = 3: 5.1.
other: absolutely irreducible.
*/


F:=Rationals();
Z:=Integers();


G<x,y>:=MatrixGroup<6,F|\[
0,1,0,0,0,0,
1,0,0,0,0,0,
1,1,-1,0,0,0,
0,0,0,0,0,1,
0,0,0,1,-1,1,
0,0,0,1,0,0]
,\[
0,0,1,0,0,0,
0,0,0,1,0,0,
0,0,0,0,1,0,
-1,0,0,0,-1,0,
-1,-1,0,0,0,0,
0,0,0,0,0,1]
>;
a:=x;b:=y;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 243 = 3^5; elem divs 1.3^5.
B1:=MatrixAlgebra(F,6)!\[
4,-2,1,-2,-2,1,
-2,4,1,1,1,-2,
1,1,4,-2,1,1,
-2,1,-2,4,1,-2,
-2,1,1,1,4,1,
1,-2,1,-2,1,4];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,6)!1;


LA1:=LatticeWithGram(MatrixAlgebra(Z,6)!B1);
LAD1:=LatticeWithGram(MatrixAlgebra(Z,6)!(3*B1^-1));